Determining the depth of surface charging layer of single Prussian blue nanoparticles with pseudocapacitive behaviors

Understanding the hybrid charge-storage mechanisms of pseudocapacitive nanomaterials holds promising keys to further improve the performance of energy storage devices. Based on the dependence of the light scattering intensity of single Prussian blue nanoparticles (PBNPs) on their oxidation state during sinusoidal potential modulation at varying frequencies, we present an electro-optical microscopic imaging approach to optically acquire the Faradaic electrochemical impedance spectroscopy (oEIS) of single PBNPs. Here we reveal typical pseudocapacitive behavior with hybrid charge-storage mechanisms depending on the modulation frequency. In the low-frequency range, the optical amplitude is inversely proportional to the square root of the frequency (∆I ∝ f−0.5; diffusion-limited process), while in the high-frequency range, it is inversely proportional to the frequency (∆I ∝ f−1; surface charging process). Because the geometry of single cuboid-shaped PBNPs can be precisely determined by scanning electron microscopy and atomic force microscopy, oEIS of single PBNPs allows the determination of the depth of the surface charging layer, revealing it to be ~2 unit cells regardless of the nanoparticle size.


Electrochemical system and optical configurations
The preparation of the electrochemical cell and electrodes were described in our previous work [2][3] . The thickness of ITO-coated glass slide was 1.1 mm (8 ohms/square, Wuhan Jinge-Solar Energy Technology Co. Ltd.). 150 μL droplet of 5 times diluted PBNPs solution was dropped on the ITO and dried in the vacuum oven (12 hours). The electrochemical measurements were performed in a 0.5 M KNO3 solution in the absence of additional redox molecules. KNO3 not only served as electrolyte to reduce IR drop, but also provided sufficiently high concentration of K + for insertion/extraction. Voltage was applied by the potentiostat (Autolab PGSTAT302N, Metrohm AG) and modulated via an external waveform function generator (RIGOL, DG1000Z). A data acquisition card (USB-6281, National Instruments) was utilized to synchronize the voltage from potentiostat and transistor-transistor logic signals from the camera.
The optical image of PBNPs was obtained with an inverted microscope (Eclipse Ti-U, Nikon), which was installed with an oil-immersed dark-field condenser (NA = 1.20−1.43), an objective lens (40x, NA = 0.6), and a 660 ± 20 nm light-emitting diode (M660L3-C1, Thorlabs) as the light source ( Supplementary Fig. 3a). The dark-field image was collected by a CCD camera (Stingray, Allied Vision Technologies). In this condition, we can visualize single PBNPs as small as 100 nm with sufficient contrast in DFM ( Supplementary Fig. 4).
A grating spectrometer (Acton Spectra Pro SP-2300, Princeton Instruments) with slit was used to capture the scattering spectra. Supplementary Fig. 3b shows the scattering spectra of PB and PW states at different voltage, with maximum scattering wavelength near 700 nm. It was believed that the scattering of PBNPs at ~700 nm was due to the Rayleigh resonant scattering, because the incident wavelength was consistent with the absorption band of Fe-Fe intervalence charge transfer 3 . When applying negative voltage, the PBNPs can be transformed into PWNPs with an obvious decrease in the scattering intensity around 700 nm.

The distribution of formal potential of PBNPs
Here, we defined the formal potential of PBNPs for better understanding the stage-of-charge of PBNPs.
The formal potential of each individual had been examined by monitoring the optical intensity as a function of potential. When recording a series of optical imaging under sweeping voltage (-300 mV − 250 mV with a sufficiently slow scan rate of 5 mV/s), the formal potential was determined at 50% between the highest and lowest intensity (Fig. 1c). It was a consequence of structural and compositional heterogeneity during synthesis.
We have accordingly selected the ones with formal potential in the range between -0.05 and 0 V for further study, which accounted for the largest portion ( Supplementary Fig. 5). The subsequent frequency modulation experiment was conducted at the offset potential of -25 mV for the largest optical amplitude.

The long-time cyclability of PBNPs during electrochemical modulation
In order to demonstrate the stability of PBNPs during the long-time experiment, PBNPs were modulated during 1000 consecutive cycles at a frequency of 1 Hz (17 minutes). The Fourier transform was utilized to extract the optical amplitude (detailed process were shown below). We exemplified the Fourier transform results by calculating the 400-500th cycles (the green box in Fig. 1f).
Supplementary Fig. 6 The fluctuation of scattering intensity during 100 consecutive cycles (400-500th cycles) at a frequency of 1 Hz, the diagram below is the amplitude after Fourier transform. 5

The distinction between Faradaic component and non-Faradaic component
The dark-field imaging method can effectively distinguish Faradaic and non-Faradaic components. When a sinusoidal voltage was applied, the scattering intensity of particles could change with the voltage, while the intensity of the ITO electrode remained unchanged ( Supplementary Fig. 7). Therefore, the redox reaction of PBNPs and charging/discharging process of electric double layer can easily be distinguished, only by selecting the rectangle region of interest (ROI) at the position of single PBNPs (red rectangle, left panel in Supplementary Fig. 7).
Supplementary Fig. 7 The scattering intensity of PBNPs and surrounding background when a sinusoidal voltage was applied.
The red area represents a single PBNP, the blue area is the background of ITO electrode. The right panel is the corresponding scattering intensity. The modulation frequency was 0.01 Hz and the amplitude was 20 mV.

The calculation of optical amplitude and phase
We then extracted the amplitude and phase of optical intensity curves. In the specific operation, it should be noted that in order to ensure the linear response of system, a small voltage was necessary. However, the small voltage would also result in a low signal-to-noise ratio of optical amplitude. Under this limitation, we optimized the amplitude of voltage at each frequency to meet the sufficient signal-to-noise ratio. In the lowfrequency region, the voltage was small, while in the high-frequency region, the voltage was required to be larger. Here, we had selected four representative frequencies, 0.1 Hz ( Supplementary Fig. 8a The Fourier transform was utilized to extract the optical amplitude and phase. Whether it was at low frequency (0.1 Hz) or high frequency (100 Hz), sufficient signal-to-noise could be obtained.

The optical transfer function (OTF) defined by oEIS
The optical transfer function (OTF) that we measured in Fig. 2a and Fig. 2b was defined as following: Here, ∆I is the optical amplitude and V is the voltage. In the subsequent processing, we had unified the amplitude of the voltage at each frequency. In other words, all the optical amplitude was calculated at the amplitude of 20 mV of voltage (Fig. 2a). The optical phase difference was the optical phase minus voltage phase (Fig. 2b).
We chose to display the OTF rather than impedance directly because of the following reasons. In order to obtain the impedance/admittance information, a first-order derivative had to be performed to the optical intensity curves (corresponding to charge quantity) to resolve the current (corresponding to charge transfer rate). Unfortunately, the first-order derivative was found to significantly increase the noise level, particularly at high-frequency range ( Supplementary Fig. 10b). It made the quantification at high frequency range challenging because the signal itself became smaller when the frequency was higher. its first order derivative. The corresponding Fourier transform results are shown in the right panel.
Although it was difficult to directly measure the optical current at each frequency, we could also build a mathematic transform to obtain the impedance/admittance (based on voltage ~ current) from OTF (charge quantity or the integration of current ~ voltage).
The amplitude and phase of the impedance can be written as: The good fitted results in Supplementary Fig. 11 show the reliability of this conversion. 9 Supplementary Fig. 11 The amplitude (|Z|, left panel) and phase (ΦZ, right panel) of impedance can be calculated from those of OTF. The red lines are fitted results and the blue dots are experimental data.
Furthermore, we used the Kramers-Kronig methods to verify the reliability of the frequency-dependent data: Supplementary Fig. 12 Bode plots of OTF amplitude (a) and phase (b). The red lines are Kramers-Kronig transform simulation results, and the blue dots are experimental data.

The linear dependence of optical intensity with ion content
In order to convincingly demonstrate that the overall scattering signal of single PBNPs (smaller than optical diffraction limit) was quantitatively dependent on its state-of-charge, we have performed the experiment by simultaneously recording the faradaic current and optical traces of one PBNP during its collision. The apparatus and methodology were adopted from our previous publication 4 . While LiCoO2 nanoparticles were studied in previous study, herein PBNPs was used instead. Briefly, a 50×50 μm 2 microelectrode was fabricated to reduce the background current. When applying a constant reduction potential (-300 mV) onto the electrode and allowing single freely-moving PBNPs in the suspension to stochastically collide onto the electrode, a transient reduction current was recorded after the collision-andstay of single PBNPs. By doing so, it was ensured that the electrode current was solely contributed by the particular nanoparticle. Since early 2000s, it has been a very powerful strategy pioneered by Lemay, Bard, Compton, and many others 5 , which is known as single nanoparticle collision/impact electrochemistry. Our contribution was to employ an optical microscopy to simultaneously record the entire collision-and-reaction process, and to quantitatively compare the optical signals with electrochemical current. As shown in Supplementary Fig. 13d, at 0.91 second, the nanoparticle collided on the electrode and therefore led to a sudden increase in the optical signal. Then, electrochemical reduction of the nanoparticle gradually increased the optical signal, indicating the gradual conversion from PB to PW. This point was confirmed by the simultaneously recorded reduction current ( Supplementary Fig. 13e). In this experiment, it was ensured that the optical signal and electrode current was from the same individual PBNP 4 . If we plotted the optical intensity as a function of quantity of injected electrons (integration of current), there was a quasilinear dependence of optical scattering signal on the state-of-charge, especially in the range between 40−80% ( Supplementary Fig. 13c).
It was necessary to clarify that, surface plasmon resonance microscopy (SPRM) rather than dark-field microscopy (DFM) was employed to obtain the results shown in Supplementary Fig. 13a. It was because a micron-sized electrode was required for this experiment to suppress the background current. The edge of microelectrode resulted in a rather high optical background in DFM. However, since both SPRM and DFM measured the optical scattering signal, we believed the monotonic dependence of optical scattering on stateof-charge remained valid in both cases.
When charging single nanoparticles, we would like to believe that the overall scattering intensity was able to quantitatively report the state-of-charge of single nanoparticle as long as it was smaller than the optical diffraction limit (~300 nm), although with the existence of the spatiotemporal heterogeneity of the electrode.
First, our results in single nanoparticle collision electrochemistry had clearly supported this point for PBNPs.
Second, nearly all individual PBNPs we investigated displayed a monotonic and smooth intensity curve during 11 electrochemical charging/discharging cycles. In addition, in our previous study on single LiCoO2 nanoparticles (~200 nm size), monotonic dependence was also observed 4 .
There were a few reasons to ensure a more straightforward and quantitative relationship between the optical signal and the ion insertion content in our study. First, because as-prepared PBNPs were around 100~300 nm and of regular cubic-shape, the diffraction effect allowed for accessing the overall change in morphology and refractive index. This scenario was in contrast to micro-particles with a size of tens of microns and irregular morphology 6 , which was superior to map the ion transport pathways with sub-particle spatial resolution by imaging the local variations of scattering from an irregular 10-micron sized LiCoO2 particle.
Second, it was believed that the scattering of PBNPs at ~700 nm (wavelength) was due to the resonant Rayleigh scattering, because the incident wavelength was consistent with the absorption band of Fe-Fe intervalence charge transfer. Since absorption of nano-sized object was less sensitive to its morphology, it would be more reliable to reflect the overall ion content within the entire nanoparticle.
In summary, the linear dependence between optical scattering intensity of single PBNPs and its state-ofcharge was reliable under these two conditions: 1) nanoparticle size was smaller than optical diffraction limit, and 2) the state-of-charge was close to 50% in our experiment.

Extracting the depth of surface charging layer
In order to further prove that the ∆I∝f -1 in high frequency region and ∆I∝f -0.5 in low frequency region, we compared the double logarithm graph. The slope of 1 indicated that the optical amplitude conformed to the following formula: Once plotting the optical amplitude as a function of the inverse of the square root of the frequency (f -0.5 ), it became clear that the curve was composed of two segments: a linear curve in the low frequency range (right part, f -0.5 ), and a parabolic curve in the high frequency range (left part, f -1 or (f -0.5 ) 2 ). It was well consistent with the proposed mechanism. In order to unbiasedly determine the corner frequency, a piecewise function (in which fcutoff is a parameter-to-be-fitted) was applied to fit the entire curve: . For the representative amplitude results shown in Fig. 2a, the corner frequency was fitted to be 0.9 Hz. So the contribution from surface charging (Qs) is calculated to be 5.9 IU. Since the total charge capacity (Qt) is 82.2 IU, the contribution percentage of pseudocapacitive behavior is 7.2%.
The single PBNP has regular cuboid geometry that is well characterized by its length (215 nm), width (200 nm) and height (215 nm). Since that PBNP exhibits a face-centered cubic structure and its crystalline size is 1.02 nm, the total number of cells in PBNP are (

The influence of the state-of-charge on oEIS
The pseudo-capacitive behavior was significantly dependent on the state-of-charge. According to this, we had conducted further experiments at different offset potentials to examine the influence of state-of-charge on the OTF as well as corner frequency (fcutoff). First, optical response as a function of sweeping potential from -300 to +250 mV (scan rate: 5 mV/s) was shown in Supplementary Fig. 17a. It revealed a formal potential of -25 mV, and a quasi-linear dependence of optical intensity with potential between -55 and 5 mV.
Then, oEIS of the same individual PBNP was measured at varying offset potentials of -10, -25, -40 mV with the same amplitude of 20 mV. The results were shown in Supplementary Fig. 18. According to the curve above, the state-of-charge under these potentials was 70% PB−30% PW, 50% PB−50% PW, 30% PB−70% PW, respectively. It was clear that the maximal optical amplitude was obtained at the formal potential of -25 mV.
Similar corner frequency was also observed in the state-of-charge range of 30%−70%, 0.7 Hz (-10 mV), 0.8 Hz (-25 mV) and 0.8 Hz (-40 mV), contributed 7.1%, 7.6% and 7.4% to the total charge storage capacity, respectively. In other words, the corner frequency (~0.8 Hz) and the depth of surface charging layer (~2 unit cells) were more or less independent on the state-of-charge, at least in the range of formal potential ± 15 mV (corresponding to state-of-charge 30~70%). We attributed such stability as the same face-centered cubic crystal structure and the similar lattice parameters between Prussian Blue (oxidized form) and Prussian White (reduced form).
At the same time, the difference in corner frequency should be more evident when an extreme state-ofcharge (such as 10% and 90%) was examined, or in another redox system with significant lattice change during cycling. Unfortunately, when we tried to apply such extreme conditions in our study, the sample tended to rapidly lose activity (fading) during consecutive and long cycling under extreme potentials. Here, R is the gas constant, T is the temperature, n is the number of electron transferred, i0 is the exchange current density, CR is the concentration of the oxidation species, Co is the concentration of the reduction 16 species, F is the Faraday constant, A is the surface area, k0 is the standard electrochemical rate constant which is independent of the potential, and α is transfer coefficient which can also be regarded as constant approximately in a narrow potential window.
Finally, Rct is considered to be concentration (or applied potential) dependent, which has the minimum value when the concentration of oxidation species and reduction species are equal. This result is in good agreement with our experimental results, the values of RNP are 0.056 (-10 mV), 0.036 (-25 mV) and 0.047 (-40 mV), respectively.

The influence of the electrical contact on oEIS
The electrical contact is critical in the electrochemical experiment. Our previous work developed a sputter-coating method by depositing an ultrathin platinum layer 7 , which investigated the influence of electrical contacts on the apparent activity of single nanoparticles. We clarify that this point remained valid, and the present oEIS offers a promising capability to quantify the contact resistance by analyzing RNP.
As an example, we collected and compared the oEIS of a very same individual before and after drying the sample in a vacuum chamber (10 -4 Pa) by 1 hr. As shown in Supplementary Fig. 19 below, the value of RNP was found to ~20 times lower after the vacuum drying. This result not only demonstrated the capability of oEIS for quantifying the contact resistance of single nanoparticles, but also provided a more feasible protocol to enhance the electrical contacts by vacuum drying (than our previous method of metal sputtering).
Supplementary Fig. 19 The normalized lOTFl of a very same individual PBNP before (red) and after (blue) vacuum drying.
The lines are the fitted results, and the dots are experimental data.
In conclusion, nanoparticle capacitor (CNP) shown in Fig. 4b described the charge storage capability of single PBNPs, including both the surface-limited charging layer and the diffusion-limited interior part. It was proportional to the volume of nanoparticles (Fig. 3g). Nanoparticle resistor (RNP) was used to describe the contact resistance at the nanoparticle-electrode junction, and charge transfer resistance that used to describe electron/ion transport within the nanoparticle. We had performed further control experiments to demonstrate this point. For example, the enhanced electrical contact (via vacuum drying), or the change in the offset potential, was able to monitor RNP as expected.

The equivalent circuit model of single PBNPs
It can be seen from the equivalent circuit (Fig. 4b) that impedance can be written as: When a voltage with amplitude of V0 was applied, the current is: .
The real and imaginary parts of the current can be written as: The amplitude and phase of the current are: According to the current, the amplitude and phase of the charge can be obtained: Finally:

Scattering intensity and equivalent radius of different PBNPs
The length and width of PBNPs can be obtained from the scanning electron microscope, and the equivalent radius is calculated by the following formula to assess the volume of PBNP: which is considered as the radius of a virtual circle with the same projection area of the nanoparticle. In the 12 nanoparticles which the length was ranging from 100 nm−300 nm, an obvious linear relationship between the scattering intensity and r 3 was observed ( Supplementary Fig. 20h). This revealed that the volume of nanoparticles (r 3 ) was proportional to the scattering intensity. So we could use the scattering intensity instead of the actual volume of particles for further statistics because the AFM imaging took a long time to capture.

The calculation of optical coefficient
Assuming that PBNP can completely convert to PWNP when a sufficiently negative potential was applied, we can see that the scattering intensity of the particle was reduced by 2095 IU in Fig. 1c. Since the following reaction occurred in the one unit cell of PBNP: ). By virtue of these above, we can establish an optical electrochemical signal conversion model to obtain the photoelectric conversion coefficient : 6.0×10 -5 IU/charge.

The equivalent circuit model of the whole entire ITO electrode
We measured the impedance of the whole electrochemical cell (two identical ITO electrodes) with the voltage amplitude of 10 mV, ranging from 0.01 Hz to 10 kHz. A 0.5 M KNO3 solution was used as electrolyte throughout the work in the absence of pH buffer. Besides, we clarified that the electrochemical measurements were performed in the absence of additional redox probe in the solution. 19 The equivalent circuit diagram (Fig. 4a) is composed of electric double layer capacitance Cdl, electrolyte solution resistance Rs, polarization resistance Rp, and Warburg impendence Zw. The total impedance is: Here, n is an adjustable constant and RW is a constant. When n is 0.5 or 1, this indicates ideal Warburg impendence or pure capacitive behavior respectively. From the simulation results, we can find there is no significant voltage drop in the frequency ranging from 0.01 Hz to 100 Hz (Fig. 4c). Resistor (Rs) described the solution resistance, which was determined to be ~42 Ohm. This value was consistent with literatures that used the same electrolyte and electrochemical cell design 2 . Double layer capacitor (Cdl) described the effect of electrical double layer, which was determined to be 2.8 μF. The capacitance density was therefore 4.7 μF/cm 2 . Polarization resistor (Rp) in series of warburg element (Zw) were frequently used to describe the interfacial Faradaic reaction involving the reduction of dissolved oxygen.
It was found that, the reduction of dissolved oxygen was responsible for the deviation of -90° phase at low frequency range as shown in Supplementary Fig. 21b. As long as the oxygen was removed from the solution by purging Argon bubbles, the deviation was reduced ( Supplementary Fig. 22).